## Abstract

Let R be a two-dimensional normal graded ring over a field of characteristic p>0. We want to describe the tight closure of (O) in the local cohomology module H_{R+}^{2}(R) using the graded module structure of H_{R+}^{2}(R). For this purpose we explore the condition that the Frobenius map F: [H_{R+}^{2} (R)]_{n}→[H_{R+}^{2} (R)]_{pn}induced on graded pieces of H_{R+}^{2}(R) is injective. This problem is treated geometrically as follows: There exists an ample fractional divisor D on X=Proj (R) such that R=R (X, D)= ⊕_{n≥0}H^{0}(XO_{X} (n D)). Then the above map is identified with the induced Frobenius on the cohomology groups[Figure not available: see fulltext.] Our interest is the case n<0, and in this case, a generalization of Tango's method for integral divisors enables us to show that F_{n} is injective if p is greater than a certain bound given explicitly by X and nD. This result is useful to study F-rationality of R. The notion of F-rational rings in characteristic p>0 is defined via tight closure and is expected to characterize rational singularities. We ask if a modulo p reduction of a rational signularity in characteristic 0 is F-rational for p≫0. Our result answers to this question affirmatively and also sheds light to behavior of F-rationality in small p.

Original language | English |
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Pages (from-to) | 301-315 |

Number of pages | 15 |

Journal | manuscripta mathematica |

Volume | 90 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1996 Dec 1 |

## ASJC Scopus subject areas

- Mathematics(all)